In a geometric series where the first term is 4 and the common ratio is 2, what is the sum of the first 5 terms? (2023)
Practice Questions
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Q1
In a geometric series where the first term is 4 and the common ratio is 2, what is the sum of the first 5 terms? (2023)
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The sum of the first n terms of a geometric series is a(1 - r^n) / (1 - r). Here, a = 4, r = 2, n = 5. So, 4(1 - 2^5) / (1 - 2) = 4(1 - 32) / -1 = 124.
Questions & Step-by-step Solutions
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Q
Q: In a geometric series where the first term is 4 and the common ratio is 2, what is the sum of the first 5 terms? (2023)
Solution: The sum of the first n terms of a geometric series is a(1 - r^n) / (1 - r). Here, a = 4, r = 2, n = 5. So, 4(1 - 2^5) / (1 - 2) = 4(1 - 32) / -1 = 124.
Steps: 14
Step 1: Identify the first term (a) of the geometric series, which is given as 4.
Step 2: Identify the common ratio (r) of the geometric series, which is given as 2.
Step 3: Identify the number of terms (n) you want to sum, which is given as 5.
Step 4: Use the formula for the sum of the first n terms of a geometric series: S_n = a(1 - r^n) / (1 - r).
Step 5: Substitute the values into the formula: S_5 = 4(1 - 2^5) / (1 - 2).
Step 6: Calculate 2^5, which is 32.
Step 7: Substitute 32 back into the equation: S_5 = 4(1 - 32) / (1 - 2).
Step 8: Simplify the expression inside the parentheses: 1 - 32 = -31.
Step 9: Substitute -31 into the equation: S_5 = 4(-31) / (1 - 2).
Step 10: Calculate 1 - 2, which is -1.
Step 11: Substitute -1 back into the equation: S_5 = 4(-31) / -1.
Step 12: Calculate 4 * -31, which is -124.
Step 13: Divide -124 by -1, which gives you 124.
Step 14: The sum of the first 5 terms of the geometric series is 124.
